The chord method for the determination of nonconvex workspaces of planar parallel manipulators

An extension of a novel optimization approach for the determination of accessible output sets of planar manipulators is presented. The optimization approach provides a general method for the determination of workspaces and has the advantage that it allows for the easy and systematic implementation of constraints acting on manipulators. The previously proposed optimization methodologies, the ray method and modified ray method, were unable to map the boundaries of nonconvex planar workspaces automatically. A new chord method is proposed in this paper, which allows for the efficient and automated mapping of most nonconvex workspaces. The chord method is illustrated by application to three degree of freedom planar parallel manipulators of varied geometry. In particular, as an extreme test and illustration of its capability, the chord method is also applied to manipulators exhibiting redundant behavior.